Strong Convergence Theorems of Common Fixed Points for a Family of Quasi--Nonexpansive Mappings

Abstract

We consider a modified Halpern type iterative algorithm for a family of quasi--nonexpansive mappings in the framework of Banach spaces. Strong convergence theorems of the purposed iterative algorithms are established.

1. Introduction

Let be a Banach space, a nonempty closed and convex subset of , and a nonlinear mapping. Recall that is nonexpansive if

(1.1)

A point is a fixed point of provided . Denote by the set of fixed points of , that is, .

One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see ([1, 2]). More precisely, take and define a contraction by

(1.2)

where is a fixed element. Banach Contraction Mapping Principle guarantees that has a unique fixed point in . It is unclear, in general, what the behavior of is as even if has a fixed point. However, in the case of having a fixed point, Browder [1] proved the following well-known strong convergence theorem.

Theorem 1 B.

Let be a bounded closed convex subset of a Hilbert space and a nonexpansive mapping on . Fix and define as for any . Then converges strongly to an element of nearest to .

Motivated by Theorem B, Halpern [3] considered the following explicit iteration:

(1.3)

and obtained the following theorem.

Theorem 1 H.

Let be a bounded closed convex subset of a Hilbert space and a nonexpansive mapping on . Define a real sequence in by , . Then the sequence defined by (1.3) converges strongly to the element of nearest to .

In [4], Lions improved the result of Halpern [3], still in Hilbert spaces, by proving the strong convergence of to a fixed point of provided that the control sequence satisfies the following conditions:

(C1)

(C2)

(C3)

It was observed that both the Halpern's and Lion's conditions on the real sequence excluded the canonical choice . This was overcome by Wittmann [5], who proved, still in Hilbert spaces, the strong convergence of to a fixed point of if satisfies the following conditions:

(C1)

(C2)

(C4)

In [6], Shioji and Takahashi extended Wittmann's results to the setting of Banach spaces under the assumptions (C1), (C2), and (C4) imposed on the control sequences . In [7], Xu remarked that the conditions (C1) and (C2) are necessary for the strong convergence of the iterative sequence defined in (1.3) for all nonexpansive self-mappings. It is well known that the iterative algorithm (1.3) is widely believed to have slow convergence because the restriction of condition (C2). Thus, to improve the rate of convergence of the iterative process (1.3), one cannot rely only on the process itself.

Recently, hybrid projection algorithms have been studied for the fixed point problems of nonlinear mappings by many authors; see, for example, [8–24]. In 2006, Martinez-Yanes and Xu [10] proposed the following modification of the Halpern iteration for a single nonexpansive mapping in a Hilbert space. To be more precise, they proved the following theorem.

Theorem 1 MYX.

Let be a real Hilbert space, a closed convex subset of , and a nonexpansive mapping such that . Assume that is such that Then the sequence defined by

(1.4)

converges strongly to

Very recently, Qin and Su [17] improved the result of Martinez-Yanes and Xu [10] from Hilbert spaces to Banach spaces. To be more precise, they proved the following theorem.

Theorem 1 QS.

Let be a uniformly convex and uniformly smooth Banach space, a nonempty closed convex subset of , and a relatively nonexpansive mapping. Assume that is a sequence in such that . Define a sequence in by the following algorithm:

(1.5)

where is the single-valued duality mapping on . If is nonempty, then converges to

In this paper, motivated by Kimura and Takahashi [8], Martinez-Yanes and Xu [10], Qin and Su [17], and Qin et al. [19], we consider a hybrid projection algorithm to modify the iterative process (1.3) to have strong convergence under condition (C1) only for a family of closed quasi--nonexpansive mappings.

2. Preliminaries

Let be a Banach space with the dual space . We denote by the normalized duality mapping from to defined by

(2.1)

where denotes the generalized duality pairing. It is well known that, if is strictly convex, then is single-valued and, if is uniformly convex, then is uniformly continuous on bounded subsets of .

We know that, if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [25] recently introduced a generalized projection operator in a Banach space , which is an analogue of the metric projection in Hilbert spaces.

A Banach space is said to be strictly convex if for all with and . The space is said to be uniformly convex if for any two sequences and in such that and . Let be the unit sphere of . Then the space is said to be smooth if

(2.2)

exists for each It is also said to be uniformly smooth if the limit is attained uniformly for . It is well known that, if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

In a smooth Banach space , we consider the functional defined by

(2.3)

Observe that, in a Hilbert space , (2.3) reduces to for all The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem:

(2.4)

The existence and uniqueness of the operator follows from some properties of the functional and the strict monotonicity of the mapping (see, e.g., [25–28]). In Hilbert spaces, It is obvious from the definition of the function that

(2.5)

Remark 2.1.

If is a reflexive, strictly convex, and smooth Banach space, then, for any , if and only if . In fact, it is sufficient to show that, if , then . From (2.5), we have . This implies From the definition of one has . Therefore, we have (see [27, 29] for more details).

Let be a nonempty closed and convex subset of and a mapping from into itself. A point is said to be an asymptotic fixed point of ([30]) if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive ([27, 31, 32]) if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied by some authors ([27, 31, 32]).

A mapping is said to be -nonexpansive ([18, 19, 24]) if for all . The mapping is said to be quasi--nonexpansive ([18, 19, 24]) if and for all and .

Remark 2.2.

The class of quasi--nonexpansive mappings is more general than the class of relatively nonexpansive mappings, which requires the strong restriction: .

Let be a uniformly convex and smooth Banach space, a nonempty, closed, and convex subset of and a closed quasi--nonexpansive mapping from into itself. Then is a closed and convex subset of .

3. Main Results

From now on, we use to denote an index set. Now, we are in a position to prove our main results.

Theorem 3.1.

Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space and a family of closed quasi--nonexpansive mappings such that . Let be a real sequence in such that . Define a sequence in in the following manner:

(3.1)

then the sequence defined by (3.1) converges strongly to .

Proof.

We first show that and are closed and convex for each . From the definitions of and , it is obvious that is closed and is closed and convex for each . We, therefore, only show that is convex for each . Indeed, note that

(3.2)

is equivalent to

(3.3)

This shows that is closed and convex for each and Therefore, we obtain that is convex for each .

Next, we show that for all . For each and , we have

(3.4)

which yields that for all and It follows that . This proves that for all .

Next, we prove that for all We prove this by induction. For we have Assume that for some . Next, we show that for the same . Since is the projection of onto we obtain that

(3.5)

Since by the induction assumption, (3.5) holds, in particular, for all . This together with the definition of implies that for all Noticing that and , one has

(3.6)

We, therefore, obtain that is nondecreasing. From Lemma 2.5, we see that

(3.7)

This shows that is bounded. It follows that the limit of exists. By the construction of , we see that and for any positive integer Notice that

(3.8)

Taking the limit as in (3.8), we get that From Lemma 2.3, one has as It follows that is a Cauchy sequence in . Since is a Banach space and is closed and convex, we can assume that as .

Finally, we show that To end this, we first show . By taking in (3.8), we have

(3.9)

From Lemma 2.3, we arrive at

(3.10)

Noticing that , we obtain

(3.11)

It follows from the assumption on and (3.9) that for each . From Lemma 2.3, we obtain

(3.12)

On the other hand, we have By the assumption on , we see that for each Since is also uniformly norm-to-norm continuous on bounded sets, we obtain that

(3.13)

On the other hand, we have

(3.14)

From (3.10)–(3.13), we obtain From the closedness of , we get

Finally, we show that From , we see that

(3.15)

Taking the limit as in (3.15), we obtain that

(3.16)

and hence by Lemma 2.4. This completes the proof.

Remark 3.2.

Comparing the hybrid projection algorithm (3.1) in Theorem 3.1 with algorithm (1.5) in Theorem QS, we remark that the set is constructed based on the set instead of for each We obtain that the sequence generated by the algorithm (3.1) is a Cauchy sequence. The proof is, therefore, different from the one presented in Qin and Su [17].

As a corollary of Theorem 3.1, for a single quasi--nonexpansive mapping, we have the following result immediately.

Corollary 3.3.

Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space and a closed quasi--nonexpansive mappings with a fixed point. Let be a real sequence in such that . Define a sequence in in the following manner:

(3.17)

then the sequence converges strongly to .

Remark 3.4.

Corollary 3.3 mainly improves Theorem of Qin and Su [17] from the class of relatively nonexpansive mappings to the class of quasi--nonexpansive mappings, which relaxes the strong restriction:

In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following result.

Corollary 3.5.

Let be a nonempty closed and convex subset of a Hilbert space and a family of closed quasi-nonexpansive mappings such that . Let be a real sequence in such that . Define a sequence in in the following manner:

(3.18)

then the sequence converges strongly to .

Remark 3.6.

Corollary 3.5 includes the corresponding result of Martinez-Yanes and Xu [10] as a special case. To be more precise, Corollary 3.5 improves Theorem 3.1 of Martinez-Yanes and Xu [10] from a single mapping to a family of mappings and from nonexpansive mappings to quasi-nonexpansive mappings, respectively.

References

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Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space.Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272

Kimura Y, Takahashi W: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space.Journal of Mathematical Analysis and Applications 2009,357(2):356–363. 10.1016/j.jmaa.2009.03.052

Plubtieng S, Ungchittrakool K: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space.Journal of Approximation Theory 2007,149(2):103–115. 10.1016/j.jat.2007.04.014